Fast MRI Reconstruction Using Deep Learning-based Compressed Sensing: A Systematic Review

Magnetic resonance imaging (MRI) has revolutionized medical imaging, providing a non-invasive and highly detailed look into the human body. However, the long acquisition times of MRI present challenges, causing patient discomfort, motion artifacts, and limiting real-time applications. To address these challenges, researchers are exploring various techniques to reduce acquisition time and improve the overall efficiency of MRI. One such technique is compressed sensing (CS), which reduces data acquisition by leveraging image sparsity in transformed spaces. In recent years, deep learning (DL) has been integrated with CS-MRI, leading to a new framework that has seen remarkable growth. DL-based CS-MRI approaches are proving to be highly effective in accelerating MR imaging without compromising image quality. This review comprehensively examines DL-based CS-MRI techniques, focusing on their role in increasing MR imaging speed. We provide a detailed analysis of each category of DL-based CS-MRI including end-to-end, unroll optimization, self-supervised, and federated learning. Our systematic review highlights significant contributions and underscores the exciting potential of DL in CS-MRI. Additionally, our systematic review efficiently summarizes key results and trends in DL-based CS-MRI including quantitative metrics, the dataset used, acceleration factors, and the progress of and research interest in DL techniques over time. Finally, we discuss potential future directions and the importance of DL-based CS-MRI in the advancement of medical imaging. To facilitate further research in this area, we provide a GitHub repository that includes up-to-date DL-based CS-MRI publications and publicly available datasets - https://github.com/mosaf/Awesome-DL-based-CS-MRI.


Introduction
Magnetic Resonance Imaging (MRI) is a highly effective medical tool that produces highquality images of soft tissues in the body.It is widely used in the lesion prognosis and diagnosis, radiation treatment planning, and follow-up examination.However, the Lancet Oncology Commission recently highlighted a severe shortage of MRI and other medical imaging technologies in low-income and middle-income countries (Hricak et al., 2021).This shortage has resulted in 2.5 million deaths worldwide.The installation of MRI scanners remains low globally, with only 7 MRI scanners per million people installed as of 2020 (Y.Liu et al., 2021).This is primarily due to the high cost of installation, operation, and maintenance.Besides, the daily throughput is limited by the long acquisition time required for each MRI scan.The long wait time reduces the number of patients that can be seen in a given day (Murali et al., 2023).Additionally, it increases the likelihood of voluntary and involuntary patient movements (Safari et al., 2023b), causing motion artifacts that affect the accuracy of the images produced.The estimated cost of motion artifacts induced by patient movements is around $364,000 per scanner annually (Slipsager et al., 2020).
MRI requires densely sampled k-space to avoid violating the Nyquist criteria, which results in longer acquisition times for high-resolution images.To reduce the imaging time, k-space can be undersampled in the phase encoding direction by increasing the spacing between kspace lines and, therefore, covering the field of view in a shorter amount of time, as illustrated in Figure 1 for a Cartesian trajectory.Compressed sensing (CS), also known as compressive sensing or compressive sampling, is a method that aims to reconstruct fullysampled k-space from undersampled k-space by exploiting the images' sparse representation in a transform domain such as Cosine and Wavelet (Haldar et al., 2010).The CS algorithms optimize a cost function (1) to iteratively reconstruct the image.However, CS algorithms are unable to completely reconstruct the high-frequency texture content of images (Ravishankar and Bresler, 2010), limiting them to acceleration factors between 2.5 and 3 (Guo et al., 2021a).In addition, these iterative techniques inevitably increase reconstruction time.
where   ∈ ℂ × → ℂ  given  >  is the encoding operator.It is composed of a coil sensitivity map, a Fourier transform, and a sampling map with the specified pattern .  ∈ ℂ  ,  ∈ ℂ  , and  are the undersampled k-space measurement, fully sampled image, and the threshold controlling the reconstruction fidelity, which is roughly equal to the expected noise level, respectively.
Parallel imaging is another approach to reduce image acquisition time.Multiple receiver coils are placed in different positions within the scanner to independently collect a portion of the k-space data (Deshmane et al., 2012;Larkman and Nunes, 2007).Each coil is most sensitive to the area closest to it (shown in Figure 2a) with the sensitivity relationship encoded in the form of sensitivity maps (shown in Figure 2b).The final image is generated by combining the individual images obtained from different coils by taking their root sum of squares (RSS) and weighting them by the corresponding sensitivity maps.et al., 2014), are illustrated for the first 15 receiver coils.The root sum of squares (RSS) indicates combined weighted receiver coil images by the corresponding sensitivity maps.
Deep learning (DL) algorithms have garnered significant interest in CS-MRI applications.These DL models have demonstrated superior reconstruction performance at a higher acceleration rate compared to traditional non-DL-based CS models (Mardani et al., 2018;Qin et al., 2018).In light of the rapid advancements in this field, we conduct a systematic review, encapsulating the latest developments in state-of-the-art DL-based CS-MRI models.Table 1 provides a concise summary of the most relevant survey articles, revealing technical differences compared to our review.We also note that significant progress has been made in DL-based CS-MRI over the last 2 years, especially with growing interest in diffusion models and the DC layer, so our review better captures current research directions.
Table 1.Related review papers from the DL-based CS-MRI.

References Year
Contributions and technical differences (Yoon et al., 2023) 2023 This review focuses on accelerated musculoskeletal MRI.It does not discuss the statistics on the quantitative metrics and acceleration rates.
(Y. Chen et al., 2022) 2022 This review details DL algorithms and provided statistics on quantitative metrics.However, our comprehensive review, in addition to those statistics, provided detailed explanations about MRI imaging, such as k-space trajectories, the implications of different sampling patterns, and parallel imaging.Our comprehensive review also discussed the clinical applications of DL-based CS-MRI and provides insights into future directions.(Bustin et al., 2020) 2020 The primary consideration is on DL-based CS-MRI for cardiac imaging.In addition to being wider in scope, our review discusses clinical applications of interest to imaging centers, provides relevant statistics, categorizes the DL method used, and lists related references.
(Xie and Li, 2022) 2022 A review about CS for medical applications.We focus on CS for MRI and categorize based on the study's training method.In addition, our comprehensive review provides details about MRI acquisition and acceleration methods.
The PubMed database was meticulously searched on February 1st, 2024, using the terms "deep learning reconstruction", "fastMRI", "unrolled optimization", "MRI reconstruction", and "MRI acceleration" for articles published from February 2024 to January 2016.Relevant studies were carefully screened by title and abstract content.Of the 873 publications identified by PubMed, 94 articles were included.Figure 3 illustrates the entire literature screening and selection process.
Figure 3: Flowchart of our study selection process.

k-space trajectories
Various trajectories have been developed in MRI for traversing k-space, including Cartesian, spiral, radial, and random trajectories.The Cartesian trajectory, as depicted in Figure 4a, consists of parallel lines equidistant from each other, with each line representing a frequency-encoding readout.The image can be reconstructed using a fast Fourier transform, but each line requires a separate RF pulse, prolonging imaging time.
The radial trajectory, first used by (Lauterbur, 1973) and shown in Figure 4b, consists of spokes radiating from the center, with an oversampling center in k-space that makes it robust to motion artifacts (Maclaren et al., 2013).However, undersampling in the azimuthal direction increases streak artifacts (Xue et al., 2012).
The spiral trajectory shown in Figure 4c was introduced to decrease the MRI acquisition time.It starts at the center of the k-space and spirals outward, similar to radial sampling, and is robust to motion artifacts.However, hardware limitations restrict imaging efficiency and increase image blurring.

Sampling patterns
To speed up the process of capturing images, a pattern or mask, denoted as  in (1), is utilized to sample k-space.Numerous sampling patterns have been introduced for data acquisition techniques including Cartesian (Safari et al., 2024), Poisson (Slavkova et al., 2023), Gaussian (C.Hu et al., 2021), radial (Terpstra et al., 2023), and spiral patterns.
Figure 5 illustrates examples of the different sampling patterns.The Cartesian pattern is typically used for brain and knee data with a Cartesian k-space trajectory while we found the Poisson and Gaussian patterns to be primarily used to train self-supervised models as described in Section 3.4.On the other hand, radial and spiral patterns are mainly used for capturing myocardial and dynamic images that are more likely to have motion artifacts.2 Deep learning

Convolutional neural network
Convolutional neural networks, also known as ConvNets, are a type of deep neural networks that are designed to analyze grid-like data such as images and speech (LeCun et al., 1995).They have gained widespread recognition after the success of AlexNet (Krizhevsky et al., 2012) and have since been used to achieve state-of-the-art performance in various medical image processing and analysis tasks.ConvNets typically consist of multiple layers, with each layer including a convolutional operator, batch normalization layers, nonlinear activation functions, and dropout layers.Nonlinear activation functions are used to facilitate the learning of complex functions.Finally, weight regularization and dropout layers are employed to mitigate overfitting.During convolution, trainable convolution kernels slide over the images to extract multiple feature maps called channels.
The network's optimum parameters are computed using a backpropagation algorithm that calculates the gradient of the cost function with respect to the parameters in each layer.Batch normalization layers are crucial in training deep ConvNets to prevent vanishing and exploding gradients.In addition, residual blocks (He et al., 2016) are a popular choice for building advanced ConvNets due to their ability to prevent gradient vanishing and facilitate smoother error surfaces (Li et al., 2018).By incorporating skip-layer connections between input and output, residual blocks can help reduce the risk of local minima.Furthermore, when combined with (batch) normalization layers, residual blocks can effectively address the problems of vanishing and exploding gradients.

U-Net
Several deep learning models with different architectures have been proposed to enhance the performance and generalization of ConvNets.Among them, U-Net, with its elegant design that utilizes skip connections between the encoder and decoder, is the best known architecture in computer vision (Ronneberger et al., 2015).It has been extensively exploited in different medical applications such as image synthesis (Han, 2017), segmentation (Dong et al., 2019), and registration (Balakrishnan et al., 2019).In recent years, U-net architectures have incorporated residual and attention layers as a backbone to increase the network's depth and improve performance.

Transformer
While ConvNets have been impressive in their results, they are limited by the local context of convolutional operations.To address this challenge, Transformers have emerged as a solution to capture global context (Dosovitskiy et al., 2020), often outperforming ConvNets.However, transformer models are fundamentally very complex and require many trainable parameters and large databases for training which can be a challenge in medical imaging.To mitigate these issue, various variations have been proposed, such as Swin Transformers (Z.Liu et al., 2021), Vision CNN-Transformer (Fang et al., 2022), and ReconFormer (P.Guo et al., 2023), which aim to reduce model size while improving or maintaining performance.

Generative adversarial network
Generative adversarial networks (GANs) are implicit methods.Thus, they do implicitly attempt to minimize likelihood function nor attempt to learn latent representation.The GAN, initially introduced in 2014, consists of two networks, generative and discriminator (Goodfellow et al., 2020).The former is trained to generate artificial data samples to approximate the target data distribution, and the latter is simultaneously trained to distinguish the artificial data from real ground truth data.Thus, the discriminators encourage the generator to generate data samples with a distribution similar to the target distribution.Variations of GANs have been developed to perform tasks including image-toimage translation, such as conditional GAN (Mirza and Osindero, 2014), StyleGAN (Karras et al., 2019), CycleGAN (Zhu et al., 2017), and Pix2Pix (Isola et al., 2017).GANs are widely used in medical imaging for tasks such as image registration, image synthesis, and MRI image reconstruction (Quan et al., 2018;Shaul et al., 2020;Yang et al., 2017).

Diffusion model
The stable diffusion model, inspired by nonequilibrium thermodynamics, aims to simplify complex and difficult-to-calculate distributions using tractable ones like normal Gaussian distributions (Sohl-Dickstein et al., 2015).This model is comprised of two steps -the forward and reverse processes (Figure 6).During the forward process, Gaussian noise is added to the initial image  0 over  steps until the data at step  becomes normal Gaussian noise   = (, ).In the reverse process, the model learns to recover the original image  0 from its noisy version given at a step  ∈ (0, ] (Chan, 2024).Stable diffusion models have been employed for medical imaging tasks such as denoising (Pan et al., 2023b), synthesis (Pan et al., 2023a), MRI distortion reduction (Safari et al., 2023b), and MRI image reconstruction (Chung and Ye, 2022;Güngör et al., 2023).

Deep learning for MRI reconstruction
The framework for DL-based CS-MRI can be divided into two main categories: data-driven and physics-driven models.Within these categories, there are two types of models: end-toend and unroll.End-to-end models take in zero-filled k-space and output fully-sampled kspace.They typically utilize a regularization term listed in Table 2 to

End-to-end models
DL end-to-end models are specifically designed to tackle the CS-MRI problem without enforcing any data acquisition model.To achieve this, these models rely on a neural network to accurately predict fully sampled data from undersampled data (B.Levac et al., 2023;Mardani et al., 2018;Yang et al., 2016).Additionally, these models are trained using various regularization techniques that help address the ill-posed inverse problem.Table 2 lists common regularization techniques utilized in these models.Dictionary learning learns a latent representation  of the input image  where the ℓ 1 norm enforces it to be sparse.
Field of expert is composed of the convolution kernel   and Φ  , which they are learned from data.
pISTA-SENSE is a projected iterative soft-thresholding algorithm that solves (1) iteratively where the transform Ψ enforces the sparsity of the reconstructed image .
Total variation enforces image smoothness by minimizing the image gradient variations.
Sparse and low-rank model minimize the nuclear norm ‖ℋ‖ * where ℋ is the Hankel matrix.
The end-to-end approach employs the same baseline models that are used for image-toimage translation, such as the U-nets (Hyun et al., 2018;Xiang et al., 2018), Swin transformers (Huang et al., 2022), and GANs (Shitrit and Riklin Raviv, 2017;Zhao et al., 2023).However, they require a larger sample dataset than unrolling CS-MRI models and tend to predict images with synthetic data.. Table 3 provides a list of selected references that used end-to-end DL models to solve DL-based CS-MRI algorithms.

Unroll optimization
Unroll CS-MRI models combine DL with a data acquisition model to solve an optimization problem iteratively.This optimization is given by Equation 2: In this equation, λ > 0 is a scalar regularization weight that balances between the data consistency and regularization terms.The data consistency term ensures that the reconstructed images x ̂ are similar to the given undersampled y Ω , thus enforcing data fidelity.The regularization term helps solve the ill-posed CS-MRI problem by imposing sparsity on the solution to guarantee the uniqueness of the reconstructed images,  ̂ (Donoho, 2006).
DL models, particularly ConvNets, are used heavily to learn the regularization term through an unroll training scheme.Similarly, the prior DL-based regularizes encode prior knowledge about the reconstructed images, such as sparsity.The unroll CS-MRI DL models have shown to outperform end-to-end methods using a network with a smaller number of trainable parameters (Aggarwal et al., 2018;Geng et al., 2023;Liu et al., 2022;Qiao et al., 2023;Qu et al., 2024).
However, the iterative nature of the unroll optimization method may increase the computation time during both training and inference steps as both the network's weights and data consistency terms are simultaneously updated.Table 4 provides a list of references that used unroll optimization models to solve DL-based CS-MRI algorithms.

Data consistency layer
The more popular approach is to train the unrolled models similarly to the end-to-end models as follows: where   is a DL model that maps the undersampled input images   to reconstruct fully sample images .The DL reconstruction and data consistency operate on the image domain and k-space domain, respectively.Although the DL part is trained without incorporating a priori information, the second term discourages the DL first part from updating the k-space parts that were not sampled (Schlemper et al., 2017).The closed form for (3) is as follows (Qin et al., 2018): This closed form is a computational layer called the DC layer at the end of DL models.The DC layer is a crucial part of the CS-MRI DL model, playing an important role in reconstructing images (Cheng et al., 2021;Korkmaz et al., 2023).The DC layer allows for a flexible design of the DL model when it is added to U-net (Murugesan et al., 2021), transformers (Wu et al., 2023), stable diffusion model (Cao et al., 2024), and so on.

Federated learning
Federated learning (FL) is a promising framework that enables the collaborative training of learning-based models across multiple institutions without the need for sharing local private data (Yang et al., 2019).The objective of FL models is to learn a global model by taking the average of local models (McMahan et al., 2017) or by ensuring the proximity of local models to the global model (Li et al., 2020).When applied to MRI image reconstruction, the FL offers unique advantages tailored to the specific challenges and requirements as follows: • MR images often contain sensitive patient information that needs to be protected.FL enables MRI models to be trained directly on the devices where the images are acquired, without the need to transmit patient data to a centralized location.This decentralization of data ensures privacy and confidentiality of patient information is maintained.
• MRI machines can vary in their hardware specifications and imaging protocols, which can lead to challenges in standardizing image reconstruction algorithms.However, FL accommodates this heterogeneity by allowing models to be trained collaboratively across different types of MRI machines, ensuring that the reconstruction algorithms are robust and adaptable to various configurations.
It is worth noting that FL models are predominantly supervised and have been developed under the end-to-end and unroll model frameworks, which have shown promising results in various applications.

Self-supervised learning
In contrast to supervised learning methods that necessitate fully sampled ground truth images, self-supervised models alleviate this requirement and are often trained using unrolling optimization techniques.The training framework of self-supervised algorithms does not mandate fully sampled ground truth images.This approach is particularly advantageous in scenarios where obtaining fully sampled data without distortions is challenging, such as myocardial perfusion with the patient's involuntary movements, which cause motion artifacts (Haji-Valizadeh et al., 2018).Self-supervised methods draw samples from the undersampled pattern  provided in (3) to generate a new pattern  and , where  =  ∪  and  =  ∖ .The former is used to train the DL model, while the latter is utilized to compute the loss (Feng et al., 2023;Heydari et al., 2024).

Assessment
In the previous sections, we covered three major DL-based CS-MRI training approaches: endto-end, unroll optimization, and the DC layer.While these methods can be applied to both supervised and self-supervised frameworks, the unroll optimization and DC layer approaches are typically used for self-supervised training.For your convenience, we summarized the advantages and disadvantages of each approach in Table 8.Most studies quantitatively compare predicted images with ground truth images.As indicated in Figure 9a, most of these studies use the structural similarity (SSIM) index or peak signal-to-noise ratio (PSNR) to compare reconstructed image  ̂ with ground truth image  as follows: where ∥⋅∥ 2 is the squared Euclidean distance,  is the number of images' voxels, and  is the maximum voxel intensity of .A higher PSNR indicates a better reconstruction.The logarithmic operator quantifies image quality that closely aligns with human perception (Safari et al., 2023a).
where   ̂ and   are the average voxel intensities in  ̂ and ,   ̂ and   are the variance, and   ̂ is the covariance between  ̂ and .The constants  1 and  2 stabilizes the division, which usually are  1 = ( 1 ) 2 and  2 = ( 2 ) 2 .SSIM ranges between -1 and 1, with the best similarity achieved by an SSIM equal to one.
The normalized mean square error (NMSE) has become more popular since 2022 to quantify the quality of reconstructed images.The NMSE is defined as Smaller NMSE values indicate better image reconstruction.However, it favors image smoothness rather than sharpness.
However, other metrics such as root mean square error, mean square error, mean absolute error, and Fréchet inception distance are rarely used, especially after a recommendation made in 2018 by Zbontar, Jure, et al. (Zbontar et al., 2018) (see Figure 8a).
Regarding the training methods, the Unroll models, including the Unroll optimization and DC layer, are the most commonly used, with a growing use of the DC layer since 2020, which is expected to continue this trend in the future.More details about these trends are illustrated in Figure 9b.
Our systematic review found that around 46% of studies used their own dataset, while fastMRI was the most frequently used public dataset.The majority of private datasets use single-coil images compared to the later's raw multi-coil raw 2D k-space data.The fastMRI dataset consists of three imaging regions: the brain, pelvis, and knee regions.Less commonly used datasets include IXI, MRIdata, Calgary, and MICCAI challenges, with only around 7%, 5%, and 3.5% usage rates, respectively (see Figure 9c).
In addition, most studies reviewed by this study tested their proposed model using acceleration factor (R) 2 ≤  < 6.The least simulated acceleration factor was  ≥ 12 and 6 ≤  < 8 with 13.4% and 7.3% of usage, respectively.The percentage of acceleration usage is summarized in Figure 9d.

Clinical evaluations
The metrics presented in Section 4.1 quantify the quality of image reconstruction, but their results may not directly correlate with clinical outcomes.Several studies have been conducted to evaluate the clinical significance of CS-MRI using DL models.For example, a study found that DL-based CS-MRI and fully sampled MRI images showed no significant differences (p-values > 0.05) in the organ-based image quality of the liver, pancreas, spleen, and kidneys, number and measured diameter of the detected lesions while reducing the imaging time by more than 85% (Herrmann et al., 2023).
Similarly, another study showed that brain MRI images accelerated up to 4 × and 14 × had sufficient image quality for diagnostic and screening purposes, respectively (Radmanesh et al., 2022).A third study found that there was no statistical significance (p-value = 0.521) between the DL-based T2-FLAIR MRI image and standard T1c MRI images in the assessment of inflammatory knee synovitis (Feuerriegel et al., 2023).These studies are consistent with another conducted to compare the diagnostic performance of DL MRI and standard MRI images in detecting knee abnormalities (Johnson et al., 2023).
A recent study found that there is no significant difference in the overall quality of MRI images generated by DL and standard fully sampled images for various MRI sequences, including T2 and diffusion-weighted imaging, for patients with prostate cancer.The study also revealed that DL MRI and standard MR images identified a similar number of Prostate Imaging Reporting and Data System (PIRADS) ≥ 3 lesions.However, the imaging time was significantly reduced by about 3.7-fold with the use of DL MRI.This study's findings suggest that DL MRI can be a viable alternative to standard MRI imaging for prostate cancer patients, as it can produce similar quality images in a significantly shorter acquisition time (Johnson et al., 2022).

Discussion
The rapid advancement of DL in the field of computer vision has led to a significant increase in the number of studies utilizing DL to solve CS-MRI, as depicted in Figure 7.In this study, we provide a comprehensive overview of DL models and training approaches, including the use of GAN and neural architecture such as U-net and vision transformers for an end-to-end approach.Recent studies have extensively used the unroll optimization and DC layers due to their advantages, allowing for the employment of smaller ConvNets.This, in turn, reduces the number of trainable parameters, minimizing the risk of overfitting.Moreover, our study also revealed that the majority of studies employ SSIM, PSNR, and NMSE metrics to measure the image quality of predicted images.
The objective of DL-based CS-MRI approaches is to decrease imaging time, thereby enhancing throughput and minimizing the chances of patient movement.Apart from this, other methods, such as super-resolution and image synthesis, can also be adopted to decrease imaging time.These approaches strive to enhance the resolution of images obtained from lower spatial resolution (Chang et al., 2024;Xie et al., 2022) and produce MRI images of high spatial resolution and signal-to-noise ratio from lower B 0 (Eidex et al., 2023).These techniques can significantly improve MRI imaging with permanent magnets having low B 0 , ultimately increasing the image quality.Portable MRI scanners widely use permanent magnets, which substantially reduce maintenance costs and make them more suitable for low-income and middle-income countries.Although these techniques can reduce the need for stronger gradient magnets and minimize patients' nerve stimulation, they were not included in this systematic review since they did not employ compressed sensing algorithms to train their models.
The coil sensitivity map, depicted in Figure 2b, is essential to consider the non-uniform sensitivity of the receiver coils.A precise sensitivity map is crucial for generating consistent and accurate MRI images and quantitative MRI maps across different hospitals.Most of the reviewed studies use the ESPIRiT algorithm (Uecker et al., 2014) provided by the Berkeley Advanced Reconstruction Toolbox (https://mrirecon.github.io/bart/) to pre-calculate the sensitivity map.However, this algorithm involves significant computation that may hinder its application in MRI-guided surgery and treatment, where rapid image reconstruction is required.We anticipate that DL-based CS-MRI techniques, which can simultaneously predict coil sensitivity maps and coil images (Feng et al., 2023;Sun et al., 2023;Z. Wang et al., 2024), will be explored further in the future, particularly for MRI-guided treatment methods such as MRI-guided adaptive radiation therapy.Furthermore, most CS-MRI methods require prior knowledge of the sampling pattern, which may not be available to the user in real clinical applications.Therefore, an approach that can handle deviations from the actual sampling mask or jointly predict the optimal sampling pattern and predict the images (Seo et al., 2022) is required.
In a recent systematic review by Hu, Mingzhe, et al., the use of language models in medical imaging was explored in detail (Hu et al., 2024).However, our comprehensive review has found that the utilization of foundational models in training DL-based CS-MRI is currently quite limited.We believe that these models could have a more significant impact on DL-based CS-MRI.By providing prior knowledge about the sampling pattern without explicitly specifying the sampling images, these models could be extremely helpful.Additionally, their prior inputs about the MRI sequence and imaging region could aid in training a model specific to that region and sequence, potentially improving the reconstruction of out-ofdistribution and out-of-region images.

Concluding remarks
The current DL-based CS-MRI models are usually trained by using 2D fastMRI datasets, limiting the spatial resolution for small lesion detection.With the advancement of MRI techniques, future research can significantly benefit from the availability of large 3D or 4D raw k-space datasets featuring abnormalities.Such datasets can enable the development of tailored 3D models, potentially allowing real-time tumor tracking during radiation therapy for patients with conditions like lung cancer.By incorporating high-dimensional datasets, it would be possible to report accurate clinical endpoints to enhance cancer prognosis.

Figure 1 :
Figure 1: The schematic diagrams of the k-space sampling pattern with a Cartesian trajectory are illustrated for (a) fully sampled data and (b) undersampled 2D MRI data in phase encoding direction with acceleration rate (R) 2 as well as (c) undersampled 3D MRI in both slice and phase encoding direction with  = 2.The yellow and red circles indicate sampled and skipped data during the data acquisition.

Figure 2 :
Figure 2: (a) Images and (b) sensitivity maps, estimated by the ESPIRiT approach (Uecker et al., 2014), are illustrated for the first 15 receiver coils.The root sum of squares (RSS) indicates combined weighted receiver coil images by the corresponding sensitivity maps.

Figure 6 :
Figure 6: Forward and reverse diffusion processes.The first row indicates data in image space, and the second row indicates the corresponding data distribution.The forward diffusion process adds Gaussian noise in  steps in a controlled way to produce normal Gaussian noise in step .The reverse diffusion process requires a model parametrized by  to learn input  0 from noise-corrupted image   in a given step .
enforce the uniqueness of the reconstructed images.On the other hand, unroll models are more complex and further classified into two types: unroll optimization and closed-form models known for the "data consistency (DC) layer."Unroll optimization models iteratively optimize the reconstruction process, while DC layer models use a closed-form equation to ensure data consistency.These models are utilized in various training scenarios, including federated learning and self-supervised training.For a comprehensive understanding of DLbased CS-MRI methods and their corresponding components and features, please refer to Figure7.

Figure 7 :
Figure 7: An overview of seven categories of DL-based CS-MRI methods.

Figure 8
Figure8shows a stacked chart of the number of publications since 2018 by category.The total number of publications has grown exponentially in recent years and interest in the DC layer method continues to increase while interest in unrolling optimization and end-to-end approaches remains consistent over time.

Figure 8 :
Figure 8: Overview of publications in DL-based CS-MRI over time.The dashed line indicates the general trend plotted using 1 + exp(0.62 × ) where t is defined in years.

Figure 9 :
Figure 9: This graph illustrates (a) The metric used in different studies, (b) the training method used in different studies, (c) the dataset used for training used in the studies, and (d) the acceleration rate of the undersampling.Abbreviations: SSIM: structural similarity index; PSNR: peak signal-to-noise ratio; NMSE: normalized mean square error; NRMSE: normalized root mean square error; MSE: mean square error; MAE: mean absolute error; RMSE: root mean square error; and FID: Fréchet inception distance.

Table 2 :
The widely used regularization terms in deep CS-MRI are summarized.

Table 3 :
Overview of supervised end-to-end models to predict the fully sampled images.

Table 4 :
Overview of supervised unroll optimization to predict the fully sampled images.

Table 5 :
Table 5 summarizes the DL-based CS-MRI trained under the DC layer framework.Overview of supervised models that used the data consistency to predict the fully sampled images.

Table 6 .
Overview of supervised models that were trained with federated learning.

Table 7 :
Overview of self-supervised models to predict the fully sampled images.